# Limit $\lim_{x \rightarrow -\infty} (1 + x^3) = -\infty.$ by epsilon delta

Prove the following using the definition of a limit: $$\lim_{x \rightarrow -\infty} (1 + x^3) = -\infty.$$ I know we have to show that for all $$x < N$$, there must be $$f(x) < M$$, but I'm not quite sure what to do from here.

Any help would be greatly appreciated,

Thank you.

• Could you, perhaps, show us what you've attempted? – Robin Goodfellow Oct 8 '14 at 16:25
• If $x < -100$, what can you say about $1 + x^3$? – Pedro M. Oct 8 '14 at 16:28

Let $M < 0$ be given, choose N such that $N < \sqrt[3]{M-1}$. For $x < N$, we have: $1 + x^3 < 1 + (M - 1) = M$. This proves the result.